\documentclass[11pt]{article}
\usepackage{graphicx}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{epstopdf}
\DeclareGraphicsRule{.tif}{png}{.png}{`convert #1 `dirname #1`/`basename #1 .tif`.png}

\textwidth = 6.5 in
\textheight = 9 in
\oddsidemargin = 0.0 in
\evensidemargin = 0.0 in
\topmargin = 0.0 in
\headheight = 0.0 in
\headsep = 0.0 in
\parskip = 0.2in
\parindent = 0.0in

\newtheorem{theorem}{Theorem}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{definition}{Definition}
\def\today{}
\def\cl{\centerline}
\def\LF{\bigskip}
\def\Lf{\smallskip}
\def\bye{\end{document}}
\def\R{\mathbb{R}}
\def\threedxm{{\tt 3D-XplorMath\ }}


\title{\vskip -40pt           % move title up
\huge About Welling Iteration 
\footnote{This file is from the 3D-XploreMath project. You can find it on the web by searching the name.}
\vskip -20pt  %reduce space between ttle and authorname.
}

\author{M.W. and R.S.P.}

\begin{document}

\maketitle

\vskip -20pt  % reduce space after authorname

\LARGE

\centerline{\includegraphics{WellingIteration.png}}
\medskip

\LF

\centerline{\bf Welling Maps and Welling Iterations}

  A {\it Welling Map\/} is certain kind of map $ W: \R^n \to \R^n$. In more detail,
a Welling Map depends on a positive integer $N \ge n+1$ and 
$N$ points $P_1,\ldots, P_N$ in $\R^n$. Given $N$ and the $P_i$ and $x \in \R^n$,
$W(x) \in \R^n$ is defined by the following:

\bigskip
\leftline{\tt Welling Map Algorithm}

\begin{enumerate}
{\tt
\item \vglue -20pt  Compute the $N$ dot-products $\hbox{dot}_i := <x,P_i>, \; i = 1,\ldots, N$.
\item Let $j$ be the first $i$ such that  $\hbox{dot}_i $ maximizes these dot-products.
\item Define $W(x) = x - P_j$.}
\end{enumerate}

 In \threedxm $n = 2$ and the integer $N$ is given by the parameter {\tt ii}, and is initially equal to $5$.
It can be changed by selecting the first item of the Settings Menu. The two coordinates of each of the 
points $P_i$ are initially set automatically by a random number generator. Each time the item {\tt Next Fractal} 
is chosen from the Action menu, the coordinates of the $P_i$ are randomized again, producing a new Welling 
Map. The random number generator is always initialized the same way when {\tt Welling Iteration} is selected 
from the Fractal and Chaos Menu, so that a sequence of experiments done with \threedxm is repeatable.
{\bf However} if {\tt Random Fractal} is chosen from the Action Menu, then the random number generator is 
first randomly initiated and then a new set of $P_i$ are chosen, allowing serendipity in the choice of Welling Map
if desired.

  A {\it Welling Iteration\/} refers to the discrete dynamical system obtained by iterating some fixed 
Welling Map. It is believed that for appropriate choices of the Welling Map, the Welling Iteration has 
interesting properties, such as a fractal attractor. The details are under active investigation and a little 
of what is known and conjectured will be discussed below. The Welling Iteration was programmed into 
\threedxm to make it possible to visualize properties of Welling Iterations and thereby, hopefully, help 
to better understand their structure.

\LF
\centerline{\bf What is Displayed}

 As soon as  {\tt Welling Iteration} is selected from the Fractal and Chaos menu (or {\tt Next Fractal} or
 {\tt Random Fractal} is selected from the Action menu) a certain array of points equi-spaced along 
 a line segment is iterated. The original collection of points is drawn on the screen, each in a different 
 color, the next hundred iterations of the points are ignored, and then after that each iteration of the 
 array is plotted (to an offscreen bitmap for speed) and every {\tt ee} iterations the offscreen bitmap is 
 copied to the screen. In each iteration, each point of the array is plotted in its original color.
 The result is that one sees on the screen the original line of colored points and a gradually growing 
 collection of iterations of these points. The interesting parts of this collection quickly become to dense 
 to see well what is going on. If one clicks the mouse on the window, then the iteration will cease. At 
 this point one can draw a small rectangle on the screen by first holding down Command, then dragging
 out the rectangle with the mouse, then releasing Command and finally releasing the mouse. The 
 small rectangle dragged out will zoom to fill the screen, and the iteration will start over (NOT continue)
 so you will be able to see what is happening in that rectangle in more detail. This can be repeated as 
 many times as desired. 

\LF
\centerline{\bf Parameter Initialization}

\begin{itemize}

\item Number of Iterations = 9000000; 
		        
\item xmin   = - 1.8;  xmax   =   1.8;  ymin   =  - 1.1;  ymax   =    1.1;                \item xCenter := (xmin+xmax)/2;   yCenter := (ymin+ymax)/2;                \item aa = 1.0;  bb = 1.0;  \hfil\break
  $(aa,bb)$ are coordinates of one end of interval of iterated points.  				  				 	
\item cc = 0.5;  dd = 0.5; \hfil\break
   $(cc,dd)$ are coordinates of other end of interval of iterated points.
   
\item tResolution = 150;  \hfil\break
The number of points interpolated between the above endpoints.  				  				
\item ee = 100;  
   For speed,  during curve iteration, drawing is done to an offscreen bitmap, and 
   the offscreen bitmap is copied to screen every ee interations.		        
\item ff = 16;   
   number of points in mouse selected interval 		        		        
\item gg = 0;    unused		      
\item hh = 0;  unused		        \item ii  = 5;   Number of WellingPoints

\end{itemize}		        
\vfil\eject 



\bye